We study the $c-approximate near neighbor problem under the continuousFr\’echet distance . Given a set of $n$ polygonal curves with $m$ vertices, aradius $\delta > 0$ and a parameter $k \leq m$ We obtain new upper bounds that providedifferent tradeoffs between approximation factor, preprocessing time, and querytime . We complement our upper bounds with matchingconditional lower bounds based on the Orthogonal Vectors Hypothesis .Interestingly, some of our lower bounds already hold for any super-constantvalue of $k$. This is achieved by proving hardness of a one-sided sparseversion of the Orthosomatic Vector problem as an intermediate problem, which webelieve to be of independent interest, which we believe to be important to our own interest . We show that anapproximation factor of $(2+\varepsilon)$ can be obtained by usingpreprocessing time and space $O(nm) and $O(\frac{m{m)^k+k) and query time in $O(‘O’O’k”k)’s previous best result for $0 < \vareppsilon\leq 1$

Author(s) : Karl Bringmann, Anne Driemel, André Nusser, Ioannis Psarros

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Keywords : bounds - time - problem - interest - approximate -

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